Significant principal components

The symbol means that the first nc significant columns of V and T are retained. The number of significant principal components, nc, which is the pseudo-rank of X, is usually unknown. Methods for estimating the number of components are discussed in Section 31.5. 

The first step in analysing a data table is to determine how many pure factors have to be estimated. Basically, there are two approaches which we recommend. One starts with a PCA or else either with OPA or SIMPLISMA. PCA yields the number of factors and the significant principal components, which are abstract factors. OPA yields the number of factors and the purest rows (or columns) (factors) in the data table. If we suspect a certain order in the spectra, we preferentially apply evolutionary techniques such as FSWEFA or HELP to detect pure zones, or zones with two or more components. 

Two significant principal components were extracted. The score plot of the two principal components is shown in Fig. 17.5, where the compounds are color-coded according to their metabolic stability (filled points represent unstable compounds,... 

The noise component of the reduced vector is the complement of to f. The squared modulus of this noise, which is the reduced squared distance (df)2 of the ith actual measurement to the point represented by its significant principal components, is... 

This method is extremely useful to detect the points that have a large noise component (outliers) and therefore are exceedingly far from the subspace of the significant principal components. 

It ranks the data in order of its significance to the observed variance, so that noise is isolated in the least significant principal components. 

A reliable way to compress a set of signals is to perform principal component analysis (PCA) (see Section VI.C) and then use a limited number of significant principal components (PCs) as new variables to describe the samples. 

In practice, the number of compounds in a series of mixtures is not always known in advance. In a complex naturally occurring mixture it may often be impossible to determine how many significant compounds are present, and even if this is known the number of significant principal components is often much less than the true number of compounds present due to spectral similarity, noise, correlations in concentrations and so on. Hence the number of columns in T can vary. The predictions as more PCs are employed will be closer to the true values. 

Example 4.7 Determining the number of significant principal components IN A DATA MATRIX... 

After PCA, die original variables (e.g. absorbances recorded at 28 wavelengths) are reduced to a number of significant principal components (e.g. three). PCA can be used as a form of variable reduction, reducing die large original dataset (recorded at... 

A set of 126 primary, secondary, and tertiary amines characterized by seven property descriptors afforded two significant principal components which described 85% of the total variance. For details, see [63]. A preliminary study of 29 amines was used for the selection of co-substrates in studies on the Willgerodt-Kindler reaction [21]. 

The simplest definition of model complexity is based on the number of terms in the model or, in other words, the model complexity is made up by the number of model variables from Ordinary Least Squares regression cpx = p), the number M of significant principal components from Principal Component Regression (cpx = M), and the number of significant latent variables from Partial Least Squares regression (cpx = M)... 

When PCA is performed on a set of compounds characterized by -+ molecular descriptors (-> physico-chemical properties, structural variables, etc.) the significant principal components are called principal properties PP because they summarize the main information of the original molecular descriptors ... 

Todeschini, R. (1997). Data Correlation, Number of Significant Principal Components and Shape of Molecules. The K Correlation Index. Anal.Chim.Acta, 348,419-430. 

Assume that q (q[Pg.62]

The presence of outlier(s) distorts the principal component directions seriously, and as the number of outliers increases, the number of significant principal components increases which contradicting the inherent linear structure of data sets. The scatter point plots of mixtures in... 

Table 17.2 Experimental design and score of the significant principal component...

A first PLS model was established from 124 reaction systems. To ensure that this set of reaction systems was not selected in such a way that the descriptor variables were correlated, a principal component analysis was made of the variation of the eight descriptors over the set. This analysis afforded eight significant principal components according to cross validation. This showed that the variance-covariance matrix of the descriptors was a full rank matrix and that there were no severe colinearities among the descriptors. 

Compounds are described by a number of molecular descriptors these are first normalized and then subjected to the —> Principal Component Analysis to reduce the dimensionality of the chemical space. The M most significant principal components are successively transformed into binary vectors where each bit corresponds to a single principal component (PC) the bit can be either 0 or 1 depending on whether the PC value is smaller or greater than the median of that component calculated on the whole library [Xue, Godden et al., 2003b]. 

CMDand CMCindicescanbeusefulintheevaluationofthediversityof —> chemical spaces in molecule library design, in selecting the optimal number of significant principal components in PCA, and in selecting the most diverse QSAR models for —> consensus analysis. 

Todeschini, R. (1997) Data correlation, number of significant principal components and shape of molecules. The K correlation index. Anal. Chim. Acta, 348, 419-430. 

Principal components analysis (see also p. 16) involves an examination of set of data as points in n-dimensional space (corresponding to n original tests) and determines (first) the direction that accounts for the biggest variability in the data (first principal component). The process is repeated until n principal components are evaluated, but not all of these are of practical importance because some may be attributable purely to experimental error. The number of significant principal components shows the number of independent properties being measured by the tests considered. 

Four significant principal components describing 85% of the variance were found. The most important variables were associated with positions 6 and 7 on the aryl ring. Lipophilic groups containing aromatic character are especially important for enantioselectivity and the length of the aliphatic side chain was also found to be important. 

Using the minimal inhibition concentration (MIC) data for nine different strains, two significant principal components could be obtained, accounting for 77.1% and 16.1%, respectively, of the data variance. The loading plot, i.e. a plot of the calculated principal components with respect to the descriptors, shows that the first component is mainly related to the seven cell-free test systems, while the second one represents the two whole-cell test results. In other words, much redundant information was obtained by measuring in nine test systems two would have been sufficient. This separation means that the potency in both test systems is governed by different physicochemical properties. 

Figure 5.32 Soft independent modeling of class analogies (SIMCA) models for different numbers of significant principal components (PCs).

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